Generic construction of efficient matrix product operators

Matrix product operators (MPOs) are at the heart of the second-generation density matrix renormalization group (DMRG) algorithm formulated in matrix product state language. We first summarize the widely known facts on MPO arithmetic and representations of single-site operators. Second, we introduce three compression methods (rescaled SVD, deparallelization, and delinearization) for MPOs and show that it is possible to construct efficient representations of arbitrary operators using MPO arithmetic and compression. As examples, we construct powers of a short-ranged spin-chain Hamiltonian, a complicated Hamiltonian of a two-dimensional system and, as proof of principle, the long-range four-body Hamiltonian from quantum chemistry.

Figure 1

(Left) Graphical representation of one component tensor $W_i$ of an MPO. (Right) Contraction corresponding to the matrix-matrix products in Eq. (4) of multiple component tensors. Labels on the legs denote the basis on this leg. Arrows indicate whether the leg is incoming or outgoing and are largely only relevant when implementing quantum number conservation. In this convention, MPOs act on matrix product states from above, with the latter having outgoing physical indices.

Figure 2

Graphical representation of MPOs for ${\widehat{s}}_3^z$ and ${\widehat{s}}_2^{+}$ on a four-site system. Numbers and letters $z$ denote the incoming and outgoing $S^z$ quantum numbers on each tensor leg. By convention, the leftmost MPO bond index $w_0$ transforms the same as the represented operator, while the rightmost MPO bond index $w_L$ always transforms as the vacuum of the system. The MPO acts on the MPS below it, mapping states with $S^z=z$ to those with $S^z=z+1$ in the second example on the second site. Dotted lines indicate the contractions which would result in the full coefficient tensor $c$ from Eq. (4).

Figure 3

Graphical representation of the fermionic creation operator $c_{3\uparrow }^{†}$ on a four-site system. Labels correspond to the fermionic particle quantum numbers.

Figure 4

Graphical representation of ${\widehat{c}}_{4,\uparrow }^{†}$ and ${\widehat{c}}_3^{†}$ in a nonhomogeneous system. Sites 1 and 3 may contain bosons, while sites 2 and 4 may contain up to four fermions. The creation operator $c_{i\left(\sigma \right)}^{†}$ is taken to create either a fermion or boson, depending on the type of the site $i$. The parity tensor $p_F={(-1)}^{n^F}$ has been used on the second site in place of the usual identity to implement fermionic anticommutation rules for $c_{4,\uparrow }^{†}$. Labels denote the $N^F$ and $N^B$ quantum numbers on the corresponding indices.

Figure 5

Product of two MPOs for the tensors on a single site $i$. The product is built the same way on all sites $i\in [1,L]$ of the system. Matching physical indices are contracted and the two left and right MPO bond indices merged into one on each side.

Figure 6

Typical singular value distribution observed during the compression of a large Hamiltonian MPO with (rescaled) and without (standard) rescaling for different system sizes. There is a sharp drop in magnitude of the singular values between those relating to components of the operator and those made redundant by the SVD. Without rescaling, the overall magnitude of singular values grows exponentially with system size, leading to numerical errors. With rescaling, all relevant singular values have magnitude independent of the system size. (Inset) Zoom-in on the relevant first 18 singular values with rescaling, which are of roughly constant magnitude independent of the system size.

Figure 7

MPO bond dimension of the representation of Eq. (14) over the length of the chain for different chain lengths, here at $h=1$, $J_x=1/2$, $J_y=1/3$, and $J_z=1/5$ to illustrate the generic case. The leftmost and rightmost bonds have dimension four, whereas the bulk bond dimension is five as in the analytical solution.

Figure 8

Maximal MPO bond dimension (left axis) and maximal block size (right axis) for the Fermi-Hubbard Hamiltonian on a cylinder with nearest-neighbor hopping and (real-space) on-site interactions. The maximal bond dimension occurs in the middle of each ring of the cylinder and is independent of the cylinder length. This is also the bond on which the size of the largest quantum number block becomes maximal if only deparallelization (DPL) compression is employed. On this problem, SVD and delinearization (DLN) result in the same bond dimensions. With either of the two, the largest quantum number block is fairly uniformly three, only dropping down to two on the inter-ring connections.

Figure 9

Maximal MPO bond dimension in the middle of the chain for the representation of the full quantum chemistry Hamiltonian in (20). The maximal bond dimension after SVD compression for even lengths behaves exactly as $w_{\max }\left(L\right)=2L^2+3L+2$, which is the optimal result. We included some data for odd lengths $L$ for completeness; the increase in bond dimension from $L=2n$ to $L=2n+1$ is consistently four with SVD. With delinearization compression, the result is exactly the same as with SVD for $L\le 16$, for larger system sizes, the optimal representation is not always found, but the scaling is still decidedly quadratic.

Figure 10

The variance $\langle {(H-E)}^2\rangle$ of the MPS approximation to the ground state of the $S=1$ Heisenberg chain, as a function of bond dimension $m$. The naive calculation of $\langle H^2\rangle -E^2$ is subject to catastrophic cancellation and cannot be obtained accurately. The properly constructed MPO representation for ${(H-E)}^2$ is well-conditioned and obtains full numerical precision. The difference $\mathrm{\Delta }=\left(\langle H^2\rangle -E^2\right)-〈{(H-E)}^2〉$ is consistently of order ${10}^{-8}$, i.e., relevant as soon as the variance becomes sufficiently accurate.